A Novel Blind Deconvolution Method with Adaptive Period Estimation Technique and Its Application to Fault Feature Enhancement of Bearing

Minimum correlated generalized Lp/Lq deconvolution (MCG-Lp/Lq-D) is an important tool to detect periodic impulses in vibration mixture. It is proved to be a more stable technique than maximum correlated kurtosis deconvolution (MCKD) to recover the fault impulse under strong noise conditions. However, MCG-Lp/Lq-D still has limitations. One of the necessary conditions for the success of MCG-Lp/Lq-D is to provide a precise period of fault. An imprecise prior period will lead to performance degradation or even failure of the method. *erefore, in this paper, a MCG-Lp/Lq-D with adaptive fault period estimation capability is proposed, adaptive minimum correlated generalized Lp/Lq deconvolution (AMCG-Lp/Lq-D). *e proposed method uses the autocorrelation function of envelope signal to estimate the fault period adaptively in each iteration and then takes the estimated period as the input parameter of MCG-Lp/Lq-D for the next iteration optimization. *e proposed method does not require precise prior fault period input, which greatly improves the fault recovery accuracy and application range of MCG-Lp/Lq-D. Eventually, simulated and experimental data verify the effectiveness and superiority of AMCG-Lp/Lq-D.


Introduction
Fault diagnosis of rotating machinery is very important for making equipment maintenance plan, preventing equipment damage, and ensuring operation safety. Failure of rotating machinery might lead to serious equipment damage and safety accidents. Many advanced fault detection methods have been proposed and applied in engineering practice. ese methods can be divided into several categories, including spectral kurtosis [1][2][3], sparse representation [4][5][6], cyclostationarity methods [7][8][9][10], signal mode decomposition methods [11][12][13], deep learning [14,15], and blind deconvolution (BD) [16][17][18][19]. BD methods can recover fault impulse adaptively from noise observation, which has been developed and promoted in fault diagnosis. e essence of BD is to adaptively construct an inverse filter to eliminate the influence of the transmission path on the collected vibration signal [20]. According to whether the prior knowledge of the period is used, BD can be divided into nonperiod dependent BD and period dependent BD. Minimum entropy deconvolution (MED) is a representative example of nonperiod dependent BD [16] and has been applied to bearing fault diagnosis by Sawalhi et al. [21]. It can recover the fault impulse in the original vibration signal by designing a filter based on the maximum kurtosis of the filtered signal. However, many researches show that MED is sensitive to large peaks in signals [18,22] and not robust enough to noise [23,24], leading to the fact that MED might not be able to recover repetitive fault-related impulses. Aiming at the deficiency of MED, Jia et al. [18] proposed sparse filtering (SF) with the generalized Lp/Lq (G-Lp/Lq) norm with p < q. Compared with kurtosis, the G-Lp/Lq norm is less sensitive to large peaks in signals [22,25], which makes SF based on the G-Lp/Lq norm minimization more robust to large impulses and noise than MED. However, when there is strong background noise in the vibration signal, SF usually needs to use a smaller p value to improve the robustness of the method. e use of smaller p values will reduce the sensing ability of the G-Lp/Lq norm to fault impulses [22,25], resulting in the failure of SF to recover any fault features. In order to overcome the defects of MED and SF simultaneously, BD using the fault impact periodic nature (period dependent BD) is developed.
Maximum correlated kurtosis deconvolution (MCKD) [24] is a typical period dependent BD. Compared with MED, MCKD uses prior knowledge of fault impulses period to enhance the periodic fault impulses submerged in noise [24]. MCKD can adaptively design the target filter by maximizing the correlated kurtosis [24] (CK) of the filtered signal. CK can not only emphasize the high kurtosis of signal, but also highlight the periodic impulse. MCKD has stronger recovery performance of the fault impulse than MED. Subsequently, multipoint optimal minimum entropy deconvolution adjusted (MOMEDA) [23] based on multi-D-norm maximization is proposed to directly solve the optimal filter. However, the case studies in [26,27] prove that both MCKD and MOMEDA are not robust enough to process strong noise signals, while BD based on the correlated generalized Lp/Lq (CG-Lp/Lq) norm minimization (minimum correlated generalized Lp/Lq deconvolution, MCG-Lp/Lq-D) shows better robustness to noise. is is because the CG-Lp/Lq norm analyzes the correlation function of the filtered signal rather than the filtered signal itself [26], which can be regarded as the extension of the G-Lp/Lq norm in the signal correlation domain. erefore, CG-Lp/Lq norm is a low-order statistic and also emphasizes the periodicity of the impulse train, which makes MCG-Lp/Lq-D more robust to random large peaks and noise in the signal, thus making it more suitable for processing strong noise signals. However, MCG-Lp/Lq-D is very demanding for the prior fault period input precision. In engineering practice, due to the influence of both rotation speed fluctuation and measurement error [28,29], the precise fault period is usually hard to be obtained through theoretical calculation. is greatly limits the application of MCG-Lp/Lq-D in engineering practice. e purpose of this study is to overcome the limitation of the dependence on the precise fault period of MCG-Lp/Lq-D.
Recently, in order to overcome the limitation of the dependence on fault period for MCKD, Miao et al. proposed improved MCKD (IMCKD) [30]. IMCKD uses the autocorrelation function of the envelope signal for filtered signal (adaptive period estimation method, APE [31][32][33]) to estimate the fault period of signal. It solves the problem that MCKD is dependent on prior period, thus improving MCKD to be a nonperiod dependent method. Inspired by IMCKD, APE can be considered to be introduced into MCG-Lp/Lq-D to construct to overcome the second limitation of MCG-Lp/Lq-D. However, when the SNR is very low, especially when there are concurrent faults in the vibration signal [34], APE might fail to detect the real fault period, which will cause MCG-Lp/Lq-D to converge to the incorrect result [34,35]. Actually, in most engineering practice, the fault period can be obtained by theoretical calculation according to the rotation speed and the component geometry parameter [6,36,37]. Although the rotation speed fluctuation and the measurement error of component geometry parameter might cause the difference between the theoretical calculation value and the real value of the fault impulse period, it is an approximate estimation of the real fault period. erefore, when APE fails under strong noise, the theoretical calculation period will be a good input of period parameters for MCG-Lp/ Lq-D. In other words, this theoretical calculation period can be used to guide APE to track the real fault period to ensure the precision of the input period used by MCG-Lp/Lq-D.
According to the above discussion, a new deconvolution method, the adaptive MCG-Lp/Lq-D (AMCG-Lp/Lq-D), is proposed in this paper. AMCG-Lp/Lq-D uses the improved APE (adaptive period estimation with constraint, APEC) method to estimate the period parameters needed by MCG- e principle of APEC is as follows. Firstly, according to the theoretical calculation period, the APEC constructs the constraint range of the period estimation which contains the real fault period. en, the APEC is used to estimate the period within this constraint range, and the estimated period is regarded as the fault period used in this iteration. If the APEC still fails within this constraint range, the theoretical calculation period is regarded as the fault period used in this iteration. Firstly, when there is an error between the input period and the real fault period, the use of APEC can make MCG-Lp/Lq-D no longer iterate with an imprecise period, which is beneficial to improve the performance of fault recovery. Because in engineering practice, the prior fault period obtained through theoretical calculation is often not precise, the use of APEC ensures that MCG-Lp/Lq-D can use precise fault period for iteration, thus expanding the practicality of MCG-Lp/Lq-D. Additionally, because the APE technique used in the IMCKD is completely adaptive, when there is strong noise interference in the signal, the estimated period might differ greatly from the real fault period. By contrast, the APEC technique can make the proposed method avoid using the period parameters which are greatly different from the real fault period during iteration.
is is beneficial to improve the computational efficiency of the proposed method. Furthermore, AMCG-Lp/Lq-D inherits the original advantages of MCG-Lp/Lq-D, including excellent robustness to noise and concurrent fault diagnosis ability. e rest of this paper is arranged as follows. Section 2 reviews the basic theory of MCG-Lp/Lq-D. In Section 3, the limitations of APE are discussed. APEC is proposed, and based on this, AMCG-Lp/Lq-D is proposed. In Sections 4 and 5, the performances of AMCG-Lp/Lq-D in the processing of vibration mixtures separate containing single fault signal and concurrent fault signal through simulated and experimental data. Finally, the conclusions are in Section 6.

Theoretical Review
MCG-Lp/Lq-D is a deconvolution method based on the CG-Lp/Lq norm. e CG-Lp/Lq norm is defined as follows: J p,q (Corr(y, T)) � N n�1 y n− T y n+T p 1/p N n�1 y n− T y n+T where Corr(y, T) is the correlation function related to filtered signals y 1 � y(1: N − T) and y 2 � y(T + 1: N). v is 2 Shock and Vibration used to denote the correlation function, v � Corr(y, T) � y 1 y 2 . T is the precise period (in sample) of the fault impact in the vibration signal. It can be calculated by T � Fs/F d , where Fs is the sampling frequency, and F d is the fault feature frequency. e definition of MCG-Lp/Lq-D is represented as where f ∈ R L is the desired filter. e output signal after deconvolution is obtained by y � x * f, ‖f‖ 2 � 1, and x ∈ R N is the input vibration mixture signal. For the convenience of numerical calculation, Hankel matrix H ∈ R (N− L+1)×L is constructed based on x. erefore, the output result after deconvolution becomes where H is defined as Due to the fact that the minimization of J p,q (v) is still a nonsmoothness problem, c � (v 2 + ε) 1/2 is used to replace v [26]. According to equations (4)-(6), the minimization problem in equation (2) can be described by the following equation: e optimization problem in (7) can be solved by the gradient descent algorithm [38].
e gradient can be expressed as e specific steps of MCG-Lp/Lq-D are as follows: Step 1: Input the precise fault period T, convert the input signal x into Hankel H, and obtain the Hankel matrixes H 1 and H 2 .
Step 3: Obtain J p,q (v) through forward propagation; calculate dJ p,q /df based on equation (8).
Step 4: Update filter coefficients by gradient descent algorithm.
Step 5: Repeat Steps 3 and 4 until the stop criteria are fulfilled.

Adaptive Period Estimation Method.
Since the fault signal of rotating machinery is usually a nonstationary amplitude modulation signal, the envelope demodulation signal is easier to judge the fault information contained in the signal than the original signal [39]. According to the definition of autocorrelation function, the position corresponding to the maximum value of the autocorrelation function for the envelope signal is the best estimation of the period for fault signal. eoretically, the periodicity of the signal can be accurately detected by using envelope signal. e autocorrelation function of the envelope is defined as follows: Shock and Vibration 3 where E x (t) is the envelope signal of time series x(t). Hilbert(·) represents the Hilbert transform. j is the unit for the imaginary part.
where A x (τ) is the envelope autocorrelation function with respect to lag τ. Figure 1 illustrates the envelope autocorrelation function of a periodic fault impulse whose real fault period is T r . According to the definition of envelope autocorrelation spectrum, when τ ≈ T r , the envelope autocorrelation function of the fault signal reaches its maximum value (except zero point) [40]. erefore, the lag τ corresponding to the maximum value of A x (τ, τ ≠ 0) is the period (in sample) of fault impulse. In order to distinguish the real period from the estimation period, the estimation period is denoted as T e .
A simulation signal with SNR � − 2 dB is used to illustrate the effectiveness of APE, where the sampling frequency of signal Fs is 10 kHz, the fault feature frequency F d is 103 Hz, and corresponding fault period is equal to 98. e simulation signal is plotted in Figure 2(a). Because the simulation signal has a high SNR, its impulse features can be clearly observed. Figure 2(b) illustrates the envelope autocorrelation function of simulation signal, where T e � 98. e period estimated in Figure 2(b) is the same as the real fault period. is shows that APE can accurately estimate the fault period of simulation signal when SNR � − 2 dB.
When SNR � − 8 dB, the waveform of the simulation signal is shown in Figure 3(a). It can be seen from Figure 3(a) that the fault impulse is disturbed by noise, and its impulse is hard to be observed. Its envelope autocorrelation function is presented in Figure 3(b), where the period detected by APE is T e � 4. is lag corresponds to a global maximum point, which suggests that APE might not be able to correctly detect the real fault period under strong noise interference. However, by careful observation of Figure 3(b), it can be found that, near the locations marked by green line and black dot, a local maximum of envelope autocorrelation function can be observed. Its corresponding lag is exactly 98, which suggests that a fully adaptive period estimation based on APE might fail. However, it is possible to obtain the real fault period if APE is used in a local range containing the real fault period.
When SNR � − 13 dB, the fault impulse is completely submerged by strong noise, and it can hardly be observed, as shown in Figure 3(c). Its envelope autocorrelation function is presented in Figure 3(d), where the period detected by APE is T e � 4 which is greatly different from the real fault period. Additionally, a local maximum is not found in local range near 98. is indicates that the real fault period cannot be detected correctly by completely relying on APE for period detection under the condition of low SNR [41].
In summary, when the SNR is high, APE can correctly estimate the period of the signal. As the noise increases, the APE will gradually fail, but APE still has possibility to detect the real fault period in a local range containing the real fault period. When the SNR is very low, APE cannot detect the real fault period even in the local range containing the real fault period. e latter two are the two defects of APE in processing the signals with strong noise, which might affect the performance of BD method based on APE (such as IMCKD). In order to overcome the above defects, this paper proposes a period detection method, APE with constraint, APEC, which uses the prior period to guide APE to perform period selection.

Adaptive Period Estimation with Constraint.
For the first defect, when the SNR of signal is relatively low, APE is interfered by other local maximums and cannot detect the real fault period. However, it can be found from Figure 3(b) that the real fault period is the maximum within a local range of the envelope autocorrelation function. erefore, it can be considered to detect the real fault period in this local range to avoid other interference maximum points, such as 4 in Figure 3(b). Based on this idea, a constraint range can be constructed based on the real fault period and its corresponding random fluctuation [28,29]. e real fault period is always hard to be obtained in engineering practices. e theoretical calculation period is usually used to replace the real fault period [36]. However, due to the influence of random fluctuation, there is a difference between the theoretical calculation period and the real fault period, with the range within 2% [28,29]. erefore, the theoretical calculation period T and the difference caused by random fluctuation can be introduced to construct the constraint range containing the real fault period in this paper. is constraint range can be used by APE to estimate the period, avoiding interference from other local maximums. e steps of constraint construction are as follows: Firstly, the theoretical calculation period T is calculated based on [6,36,37]. Secondly, based on the random fluctuation range in [28,29], in order to extend the application range of the proposed method, 5% random fluctuation is used in this paper to construct this constraint, irdly, APE is used to estimate the period within the constraint range. For the second defect, when the SNR of signal is extremely low, APE cannot find the local maximum value within the constraint range containing the real fault period, as shown in Figure 3 Figure 1: Envelope autocorrelation spectrum of signal.
is assigned to the estimation period T e to ensure that APE can still give a period approximate to the real fault period. By improving the above two defects, the proposed APEC can ensure that the estimation period T e will not deviate from the real fault period T r to a large range even in the case of the failure of APE.

Adaptive Minimum Correlated Generalized Lp/Lq
Deconvolution. It can be known from IMCKD that the introduction of period estimation technology can improve the performance of MCKD. Inspired by IMCKD, this paper introduces APEC into MCG-Lp/Lq-D, thus proposing AMCG-Lp/Lq-D.
e proposed method is a novel fault    Shock and Vibration 5 detection technique, which can adaptively find the real fault period in each iteration by APEC to realize deconvolution algorithm. e flowchart of the proposed method is given in Figure 4. e specific implementation steps of AMCG-Lp/ Lq-D are as follows: Step 1: Input the theoretical calculation period T, select the maximum count of iteration n and the length of filter L, and initialize the filter coefficients of AMCG- Step 2: Construct the constraint T C containing the real fault period to obtain APEC, according to T.
Step 3: Obtain the estimation period T e by APEC. Format the Hankel matrix H, according to T e , and obtain the Hankel matrixes H 1 and H 2 .
Step 4: Obtain J p,q (Corr(x, T)) by forward propagation; subsequently calculate the gradient term dJ p,q /df based on equation (8).
Step 5: Update the filter coefficients f by gradient descent algorithm.
Step 6: Calculate the output signal filtered by the updated filter, y (i) � x T * f.
Step 7: Repeat Steps 3 to 6 until the maximum number of iterations is reached.
Considering that the length of filter will also affect the performance of the proposed method, a recommended selection strategy for filter length is proposed to ensure that the filter can completely cover the whole fault resonance frequency band. Equation (11) should be satisfied when selecting the filter length.
where f RS is the resonance frequency. Due to the iteration process of the proposed method, too long filter length will significantly increase the computational complexity, whereas too short filter length will cause performance deterioration of the proposed method. erefore, in order to avoid the above two extreme results, a tradeoff must be made when choosing the filter length. It should be noted that in order to ensure the fairness and effectiveness of the proposed method compared with other methods, in the subsequent simulation analysis section and experimental analysis section, all the BD methods used for processing vibration signals use filter length of 100, and the maximum number of iterations is 400. Additionally, p � 1 and q � 2 are used by both AMCG-Lp/Lq-D and MCG-Lp/Lq-D. e sampling frequencies of all measured signals are all 10 kHz; the sampling time is 1 s.

Simulation Case Studies
Two bearing simulation signals that separately contain single fault and concurrent fault are used to test the performance of the proposed method. As a comparison, IMCKD, MCKD, and MCG-Lp/Lq-D are also used to process these simulated vibrations.

Single-Fault Simulation Signal
Test. According to equation (11) [24,42] a set of impacts is designed: where A represents the amplitude of the simulated, and h(t) represents the unit step function. e real fault period is T r � Fs/F d . τ i denotes the effects of random slippage, which usually accounts for 1-2% of T r . β denotes the structure damper coefficient. Table 1 shows the parameters of simulation bearing fault signal. A Gaussian noise whose SNR � − 12 dB is added to the simulation signal S(t). e single fault simulation vibration signal is shown in Figure 5(a), and its envelope spectrum is shown in Figure 5(b). e red dotted lines represent harmonics corresponding to F d � 113Hz.
In order to prove the robustness of the proposed method to the input prior period. eir prior fault feature frequency will be set as 108 Hz and 116 Hz that deviate from the real fault feature frequency, and the corresponding fault period will be 92.6 and 86.2, respectively. e deviation fluctuation between these two periods and the real period is within 5%. MCG-Lp/Lq-D and MCKD use the same prior period input as AMCG-Lp/Lq-D. e signal in Figure 5 is processed by AMCG-Lp/Lq-D, MCG-Lp/Lq-D, MCKD, and IMCKD, respectively. e results are shown in Figures 6-8.
When the input fault feature frequency is set to 116 Hz higher than the real fault feature frequency, it can be observed that only the proposed MCG-Lp/Lq-D can detect the fault features of bearing and greatly enhance its amplitude, as shown in Figures 5(b), 6(b), 6(d) and 6(f ). e variation curve of the period obtained by APEC during the iteration process is plotted in Figure 9(a). It can be found that APEC converges to T e � 88 ≈ 10000/113 � T r at the red dot mark, which indicates that even when the prior period is imprecise, AMCG-Lp/Lq-D can adaptively correct the period through APEC and eventually converge to the real period. However, neither MCG-Lp/Lq-D nor MCKD can detect the fault features, which indicates that these two methods have higher requirements on the precision of the prior period. is is also the main limitation of MCG-Lp/Lq-D and MCKD in engineering practice.
When the input fault feature frequency is set to 108 Hz lower than the real fault feature frequency, the proposed AMCG-Lp/Lq-D still has a good fault feature recovery performance by comparing envelope spectra in Figures 5(b), 8(d), and 8(f ). By observing Figure 9(b), APEC converges from the red dot mark to T e � 89 ≈ 10000/113 � T r , which again proves the superiority of AMCG-Lp/Lq-D. Although MCG-Lp/Lq-D also detects the real fault feature frequency, its recovered feature amplitude is significantly smaller than AMCG-Lp/Lq-D. 6 Shock and Vibration By contrast, as shown in Figures 7(a) and 7(b), IMCKD cannot detect the fault features under the condition of strong noise. e variation of the estimation period for IMCKD during the iteration is shown in Figure 9(c). It can be seen that although IMCKD can also update the fault period, it cannot converge to the real fault period. is indicates that APE has a large defect in processing strong noise signals.
In order to fully study the computational complexity of AMCG-Lp/Lq-D, the computational efficiency of the four BDs is given in Table 2. It is worth noting that the calculation times shown in Table 2 are the average of each method running 30 times. e equipment used is 2.60 GHz Intel Core i7-6700HQ. e software version used is Matlab R2019 B. It can be seen that, due to the increase of APEC    Shock and Vibration technology, AMCG-Lp/Lq-D does take more time to complete the iteration than MCG-Lp/Lq-D. However, there is no doubt that only the proposed method can successfully estimate the real fault period and achieve the best fault feature enhancement effect among four methods. is indicates that the proposed method can obtain better fault detection results without significantly increasing computational complexity.
In summary, IMCKD, MCKD, and MCG-Lp/Lq-D cannot obtain satisfactory results when the SNR of simulation signal is low and the prior period is different from the real fault period. By comparing Figures 8(a) and 8(e), it can be seen that MCKD is sensitive to strong noise, resulting in its convergence to a single large peak. However, MCG-Lp/ Lq-D has a pretty good antinoise capability, which is the reason that MCG-Lp/Lq-D can still detect fault features  weakly even without precise prior period, as shown in Figure 8(d). AMCG-Lp/Lq-D shows very strong fault features detection capability under the above two prior periods, and the period of its filtered signal converges to the real fault period. is indicates that the proposed APEC has stable period estimation capability under the condition of strong noise, which ensures that the deconvolution method converges to the correct result.

Concurrent Fault Simulation Signal Test.
A concurrent simulation signal is used to verify the superiority of AMCG-Lp/Lq-D. e concurrent simulation vibration signal is made by adding a set of new single fault simulation impacts to the simulation signal in Figure 5(a). e parameters of the newly added simulation impacts are listed in Table 3. e timedomain waveforms of the simulation signal containing concurrent fault and its envelope spectrum are shown in Figure 10. In Figure 10(b), the red dotted line corresponds to F d � 91 Hz, while the yellow line corresponds to F d � 112 Hz. IMCKD, MCKD, and MCG-Lp/Lq-D are also used to deal with this signal. e results of these three methods are compared with that of the proposed AMCG-Lp/ Lq-D.
Firstly, the fault with feature frequency of 91 Hz is detected. e prior period is set as 105.3 and its corresponding frequency is 95 Hz. e concurrent fault simulation vibration signal is processed by AMCG-Lp/Lq-D, MCG-Lp/Lq-D, and MCKD, and the results are shown in Figure 11. In Figure 11(b), it can be observed that AMCG-Lp/Lq-D accurately identifies the fault features corresponding to 91 Hz in the concurrent fault. In the meanwhile, according to Figure 12(a), after about nine iterations, APEC converges to T e � 110 ≈ 10000/91 � T r at the red dot mark. Although this fault is also identified by MCG-Lp/Lq-D, its recovery performance of fault features is not ideal, as shown in Figure 11(d). MCKD converges to the single peaks in the signal and completely fails to recover the fault features submerged in the noise, as shown in Figure 11(f). is case demonstrates the importance of precise prior period to the performance of MCKD and MCG-Lp/Lq-D. However, the proposed method is very robust. Even if the input prior period parameters are imprecise, AMCG-Lp/Lq-D can still obtain good results.
Secondly, the fault with feature frequency of 113 Hz is detected. e prior period is set as 87 and its corresponding frequency is 115 Hz. e processing results of AMCG-Lp/ Lq-D, MCG-Lp/Lq-D, and MCKD are shown in Figure 13. It can be found that only the proposed AMCG-Lp/Lq-D can clearly find the fault features corresponding to 113 Hz in the composite fault, whereas the other two methods cannot detect the fault features. e iteration process of APEC for AMCG-Lp/Lq-D is shown in Figure 12 Since the concurrent fault has two different fault periods, IMCKD theoretically cannot detect two different periods at the same time. is is the main reason for the failure of IMCKD in this concurrent fault diagnosis task, as shown in Figure 14(b). e period variation of the filtering signal during its iteration is shown in Figure 12(c). Additionally, since APE performs global period detection, the estimated range of period fluctuation is too large, resulting in that IMCKD fails to converge, as shown in Figure 12(c). erefore, IMCKD cannot detect any faults. By contrast, the results in Figures 9(a), 9(b), 12(a), and 12(b) show that the estimation period obtained by APEC rapidly converges to a stable value after a certain number of iterations. is makes Shock and Vibration 9 AMCG-Lp/Lq-D have a strong capability to detect the real fault period when dealing with the signal containing single or concurrent faults. e simulation case study shows that the proposed AMCG-Lp/Lq-D exhibits very strong robustness under the condition of strong noise and when there is a difference between the given prior period and the real period. However, none of the other three methods can achieve good results. IMCKD completely modifies MCKD to a nonperiod dependent method, which makes IMCKD unable to realize concurrent fault detection. Additionally, due to the influence of strong noise, its period estimation method, APE, cannot find the fault period accurately, which eventually leads to IMCKD unable to converge.

Experimental Case Studies
In this section, a group of single fault experimental data and a group of concurrent fault experimental data are used to test the proposed method to verify its application value in engineering practice. Both sets of experimental data were obtained through the test rig shown in Figure 15(a). e main components of the test rig are motor, testing wheelset, driving wheels, and axle box. e bearings tested are axle box bearings of high-speed railway vehicles, which belong to double-row tapered roller bearings. Table 4 demonstrates the geometric dimensions of the bearings used, where d P (mm) is the pitch diameter, d R (mm) is the roller-ball diameter, Z is the number of balls, and θ(rad) is the contact angle of balls. e test wheelset is driven by the driving wheel at the bottom and supported by the axle box bearing. Figure 15(b) shows the installation position on the axle box of the accelerometer used to collect the vibration mixture signals. An ZW9609A-18SN7068 accelerometer for collecting vibration signals is mounted on the axle box, as shown in Figure 15(b). Figures 15(c) and 15(d) separately demonstrate the obvious outer-race faults and rolling element faults of the two tested bearings. It is worth noting that the bearing with rolling element fault also has obvious outer-race fault. Both sets of experimental data were obtained at speeds close to 100 km/h (rotation frequency is 10.29 Hz). e feature frequencies corresponding to the outer-race fault and the rolling element fault are 83.33 Hz and 67.86 Hz, respectively.
It must be noted that the test rig experiment has the conditions to precisely collect the relevant parameters used in precise calculation of the real fault feature frequency. However, in engineering practices, the acquisition of some parameters is restricted by the testing environment, resulting   in that they cannot be precisely obtained. In order to verify the superiority of the proposed method in engineering practices, in the case studies of this section, the fault feature frequency will be calculated by using some values deviating from the real speed instead of using the real speed. is is used to simulate the possible speed measurement error in practical engineering.

Single-Fault Experimental Data
Test. e single outerrace fault experimental vibration signal and its envelope spectrum of the bearing outer-race fault are shown in Figure 16. e red dotted line in Figure 16(b) marks the feature frequency of the bearing outer-race fault. When the operation speed is 106 km/h (the corresponding shaft rotation frequency is 10.87 Hz), the input prior period is set as 113.7, and the corresponding frequency of this period is 88 Hz. Figures 17 and 18 show the processing results of AMCG-Lp/Lq-D, MCG-Lp/Lq-D, MCKD, and IMCKD. It can be found that when the prior fault feature frequency is set to 88 Hz higher than the real fault feature frequency, the proposed AMCG-Lp/Lq-D precisely detects the real fault features and greatly improves its amplitude. is can be observed by making comparison among the envelope spectra before and after processing in Figures 16(b), 17(b), 17(d), 17(f ), and 18(b). All other methods failed. e APEC variation of the proposed method in the iteration process is shown in Figure 19(a). It can be observed that it converges to T e � 119 ≈ 10000/83.33 � T r . Although MCG-Lp/Lq-D also detects the fault features, its recovery performance is far from that of the proposed AMCG-Lp/Lq-D. By contrast, MCKD and IMCKD are completely unable to detect the fault features.
e case study shows that the performance of fault feature detection for MCG-Lp/Lq-D and MCKD will be seriously affected if the precise prior period of bearing fault cannot be provided under strong noise interference. Due to the defects of APE, IMCKD cannot accurately detect the real fault period, resulting in that the algorithm fails to converge, as shown in Figure 19(b). By contrast, the APEC in the proposed AMCG-Lp/Lq-D can approach the real fault period continually in the iteration and finally makes AMCG-Lp/Lq-D converge to the real period, so as to detect and enhance the fault features. is conclusion is the same as that obtained in the simulation analysis of bearing single fault in Section 4.1.
In order to test whether the computational complexity of the proposed method in engineering practice is consistent with the conclusion, related to computational complexity, obtained in the simulated case study, the computational efficiency of the four methods is given in Table 5. AMCG-Lp/Lq-D, MCG-Lp/Lq-D, MCKD, and IMCKD consume 20.03 s, 14.47 s, 13.12 s, and 24.52 s, respectively, for iterative calculation. It can be seen that the calculation time of AMCG-Lp/Lq-D and IMCKD is indeed longer than that of MCG-Lp/Lq-D and MCKD due to the addition of adaptive period estimation technology. However, only AMCG-Lp/Lq-D accurately estimates the real fault period and finally detected the outer-race fault features through 22 iterations when the input period deviates significantly from the real fault period. By contrast, IMCKD takes longer calculation time, and its period detection effect is not ideal because of serious noise interference.
is further indicates that APEC used in the proposed method has better period tracking ability in engineering practice than APE used in IMCKD, especially in the case of strong background noise.    extract the bearing outer-race fault features from the concurrent fault and enhance its amplitude. In addition, as shown in Figure 23           Shock and Vibration 21

Conclusions
In this paper, a BD independent of the precise prior fault period is proposed, AMCG-Lp/Lq-D. It can still detect the periodic impulse associated with the fault from the vibration signals with strong noise even when the input prior period is not precise. erefore, the proposed method can be considered as an improvement to significantly expand the application value of MCG-Lp/Lq-D, because in engineering practice, the fault period obtained through theoretical calculation is often not precise. It overcomes the problem of the dependence of MCG-Lp/ Lq-D on the precise prior period, which allows the proposed method to have a wider range of applications than MCG-Lp/Lq-D. Simulation and experimental case studies verify three advantages of the proposed method. Firstly, the proposed method can gradually estimate the real fault period in iteration without significantly increasing the computational complexity. is ensures that the proposed method will not always use an imprecise fault period to iterate, thus guaranteeing the performance of fault feature extraction for the proposed method, which is unable to be achieved by MCG-Lp/Lq-D. Secondarily, compared with the fully adaptive period estimation technique used by IMCKD, the APEC technique used by AMCG-Lp/Lq-D still has better ability of precise fault period detection even under strong noise interference. irdly, the proposed method still has the ability to diagnose concurrent faults. Based on the fault feature detection performance of the proposed method under very low SNR conditions, the proposed method is more suitable for bearing fault diagnosis tasks in engineering practice than MCG-Lp/Lq-D.

Data Availability
e experimental data are based on the CRRC project and are confidential.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

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